- Email: [email protected]
Some properties of the topological bond order
Chemical Physics Letters 407 (2005) 73–77 www.elsevier.com/locate/cplett
Some properties of the topological bond order Ivan Gutman a
a,*
, Slavko Radenkovic´ a, Boris Furtula a, Haruo Hosoya
b
Faculty of Science, University of Kragujevac, P.O. Box 60, YU-34000 Kragujevac, Serbia & Montenegro b Department of Information Sciences, Ochanomizu University, Bunkyo-ku, Tokyo 112–8610, Japan Received 4 February 2005; in final form 17 February 2005 Available online 7 April 2005
Abstract The topological bond order is a bond-order-like quantity, put forward in the 1970s. It is defined as pTrs ¼ ZðGrs Þ=ZðGÞ, where G is the molecular graph, Grs is obtained from G by deleting from it the adjacent vertices labelled by r and s, and Z stands for the respective topological (Hosoya) index. Because no easy way for the calculation of pTrs is known, its properties were studied only to a limited degree. We now introduce a modified topological bond order, ~ pTrs , that can (easily) be calculated from the eigenvalues of G and Grs. T T For acyclic systems, ~prs ¼ prs . In the case of polycyclic systems a reasonably accurate linear correlation exists between ~ pTrs and pTrs . Thus, by studying ~pTrs the main properties of pTrs can be established. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction In the topological theory of conjugated molecules [1,2] several bond-order-like quantities are considered. Of these, the best known are the Pauling et al. [3], the Coulson [4], and the Ham and Ruedenberg [5] bond orders. In the 1970s, one of the present authors proposed [6] the topological bond order, defined by means of the topological index.1 Thus, the topological bond order pertaining to the bond between atoms r and s is defined as pTrs ¼
ZðGrs Þ ; ZðGÞ
ð1Þ
where G is the molecular graph, Grs is the subgraph obtained from G by deleting from it the vertices corresponding to the atoms r and s, and Z stands for the
respective topological index. The topological index is defined as [7] X ZðGÞ ¼ mðG; kÞ; ð2Þ kP0
where m(G,0) = 1, m(G,1) = number of edges of the graph G, and m(G,k), k P 2, is the number of selections of k mutually independent (that is, mutually nontouching) edges in the graph G. Some fundamental properties of the topological bond order were established in two papers published in the 1970s [6,8], where also its relations to the Coulson and Pauling bond orders were analyzed. After that not much research in this area was done, as seen from the reviews [9,10]. An object, closely related to Z is the matching polynomial, defined as [11] X aðG; kÞ ¼ ð1Þk mðG; kÞkn2k ; ð3Þ kP0
*
Corresponding author. Fax: +38 134 335 040. E-mail address: [email protected] (I. Gutman). 1 The quantity Z, put forward in [7] and named Ôtopological indexÕ, is nowadays in the chemical literature usually referred to as the ÔHosoya indexÕ. We, nevertheless, call it by its original name. 0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.02.120
where n is the number of vertices of G. Details of the theory of the topological index and matching polynomial can be found in the books [12] and [13], respectively. Comparative studies of various bond orders
74
I. Gutman et al. / Chemical Physics Letters 407 (2005) 73–77
(including the topological bond order) were reported in [14,15]. Combining (2) and (3) one immediately obtains ð4Þ ZðGÞ ¼ in aðG; iÞ; pffiffiffiffiffiffiffi where i ¼ 1. Furthermore, if l1,l2, . . . ,ln are the zeros of the matching polynomial, then vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uY i u n h 2 ð5Þ 1 þ ðlj Þ : ZðGÞ ¼ t
Recall that DZ ¼ Z~ Z is considered as an Ôaromaticity indexÕ [10,16]. In non-alternant conjugated systems, the right-hand side of (8) may be complex-valued. Indeed, if we write the characteristic polynomial in the form X ak knk /ðG; kÞ ¼ kP0
then
j¼1
Formulas (4) and (5) are applicable to any (molecular) graph G. If G is acyclic, then the matching polynomial a(G,k) coincides with the characteristic polynomial / (G,k) [13], implying ZðGÞ ¼ in /ðG; iÞ
ð6Þ
and vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uY i u n h 2 ZðGÞ ¼ t 1 þ ðkj Þ ;
ð7Þ
j¼1
where k1,k2, . . . ,kn are the respective graph eigenvalues. In contrast to the zeros of the matching polynomial, graph eigenvalues are easily computed (using the powerful methods of linear algebra). Therefore, in contrast to (5), formula (7) provides a convenient route for the computation of the topological index. Combined with (1), formula (7) makes it possible to easily evaluate the topological bond orders of acyclic p-electron systems. Unfortunately, this graph-eigenvalue-based method for computing pTrs is not applicable in the case of cycle-containing conjugated molecules, which, from both experimental and theoretical points of view, are the most interesting and most frequently examined p-electron systems. In order to evade the computational difficulties encountered in the study of the topological bond orders of cyclic molecules, we consider an approximation whose essence is that the right-hand sides of Eqs. (6) and (7) are extended to all (molecular) graphs. The modified topological index Z~ thus obtained, namely ~ ZðGÞ ¼ in /ðG; iÞ
ð8Þ
was also investigated already in the 1970s [16], when also the expression vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uY i u n h 2 ~ ð9Þ ZðGÞ ¼t 1 þ ðkj Þ j¼1
~ rs Þ ZðG ~ ZðGÞ
was, so far, not investigated in due detail.
X X k k ð1Þ a2k i ð1Þ a2kþ1 kP0
ð10Þ
kP0
which is real-valued provided all odd coefficients (a2k + 1) are equal to zero. The fact that some odd coefficients of the characteristic polynomials of the molecular graphs of non-alternant hydrocarbons differ from zero, and that for alternant species all these coefficients are equal to zero, is a well-known consequence of the pairing theorem [1,2,12]. In what follows we restrict our considerations to alternant hydrocarbons, in which case this difficulty does not occur.
2. Relations between p~Trs and pTrs For acyclic systems, the topological bond order (1) and its modified version (10) coincide. In the case of polycyclic (molecular) graphs the relation between the two bond orders is much more complicated. Let G be a graph and C, C 0 , C 0 0 , . . . be its cycles. Let |C| denote the size of the cycle C and let G C be the subgraph obtained from G by deleting from it all vertices of C. Two cycles C and C 0 are said to be disjoint if they have no common vertex. For an example see Fig. 1. Let, as before, /(G) = /(G,k) and a(G) = a(G,k) denote the characteristic and matching polynomials of G. Then [13,18,19]: /ðGÞ ¼ aðGÞ 2 8
X
aðG CÞ þ 4
aðGÞ ¼ /ðGÞ þ 2
aðG C C 0 Þ
C;C 0
C
X
X
aðG C C 0 C 00 Þ þ
C;C 0 ;C 00
þ8
was established [17]. However, the analogous modified topological bond order, viz. ~ pTrs ¼
in /ðG; iÞ ¼
X
X
/ðG CÞ þ 4
C
X C;C
/ðG C C 0 Þ
0
0
/ðG C C C 00 Þ þ
C;C 0 ;C 00
where the summations go over all cycles, over all pairs of disjoint cycles, over all triples of mutually disjoint cycles, etc., of the graph G. In the case of alternant hydrocarbons, when all |C|-values are even, by means of (4) and (8) we get:
I. Gutman et al. / Chemical Physics Letters 407 (2005) 73–77
~ ZðGÞ¼ZðGÞþ2½ZðGC 1 ÞþZðGC 2 ÞþZðGC 3 Þ þZðGC 4 ÞþZðGC 5 ÞþZðGC 6 Þ
6 2 3
1
5
7
4
8
þ4ZðGC 1 C 3 Þ; ~ ZðG12 Þ¼ZðG12 Þþ2½ZðG12 C 2 ÞþZðG12 C 3 ÞþZðG12 C 5 Þ; ~ ~ ~ ~ ZðGÞ¼ ZðGÞ2½ ZðGC 1 Þþ ZðGC 2 Þþ ZðGC 3 Þ
9
14
10
12 11
13
75
G
G12
C1
C2
~ ~ ~ þ ZðGC 4 Þþ ZðGC 5 Þþ ZðGC 6 Þ ~ þ4ZðGC 1 C 3 Þ; ~ 12 Þ2 ZðG ~ 12 C 2 Þþ ZðG ~ 12 C 3 Þþ ZðG ~ 12 C 5 Þ : ZðG12 Þ¼ ZðG
Bearing in mind that Z(G C1) = Z(G C3) = 44, Z(G C2) = 25, Z(G C4) = Z(G C5) = 5, Z(G C6) = 1, Z(G C1 C3) = 2, and Z(G12 C2) = 10, Z(G12 C3) = 12, Z(G C5) = 2, we get
C3
~pT12 ¼
C4
¼
C6
C5
ZðG12 Þ þ 2ð10 þ 12 þ 2Þ ZðGÞ þ 2ð44 þ 25 þ 44 þ 5 þ 5 þ 1Þ þ 4ð2Þ ZðG12 Þ þ 48 ; ZðGÞ þ 256
i.e. ~pT12 ¼
G-C2
G-C1
G12-C3
G12-C5
Fig. 1. The molecular graph G of phenanthrene and of its vertexdeleted subgraph G12. The cycles contained in G are C1,C2, . . . ,C6 and are indicated by heavy lines. The only two disjoint cycles in G are C1 and C3. The graph G does not possess triples, quadruples, etc., of mutually disjoint cycles. The subgraph G12 contains only the cycles C2, C3, C5. Some typical subgraphs of the form G C and Grs C are also shown.
pT12 þ 48=ZðGÞ : 1 þ 256=ZðGÞ
~ C 1 Þ ¼ ZðG ~ C 3 Þ ¼ 48, Bearing in mind that ZðG ~ ~ ~ ZðG C 2 Þ ¼ 25, ZðG C 4 Þ ¼ ZðG C 5 Þ ¼ 5, ZðG ~ ~ C 6 Þ ¼ 1, ZðG C 1 C 3 Þ ¼ 2, and ZðG12 C 2 Þ ¼ 10, ~ C 3 Þ ¼ 12, ZðG ~ C 5 Þ ¼ 2, we get ZðG ~ 12 Þ 2ð10 þ 12 þ 2Þ ZðG ~ ZðGÞ 2ð48 þ 25 þ 48 þ 5 þ 5 þ 1Þ þ 4ð2Þ ~ 12 Þ 48 ZðG ¼ ; ~ ZðGÞ 256
pT12 ¼
i.e. ~ ZðGÞ ¼ ZðGÞ 2 þ4
X
X
ð1Þ
jCj=2
pT12 ¼ ZðG CÞ
C
ð1Þ
In the general case ðjCjþjC 0 jÞ=2
ZðG C C 0 Þ
C;C 0
8
X
0
ð1ÞðjCjþjC jþjC
00
jÞ=2
C;C 0 ;C 00
ZðG C C 0 C 00 Þ þ X jCj=2 ~ ~ ZðGÞ ¼ ZðGÞ þ2 ZðG CÞ ð1Þ þ4
X
C 0
~ C C0Þ ð1ÞðjCjþjC jÞ=2 ZðG
C;C 0
þ8
X
~ ~pT12 48=ZðGÞ : ~ 1 256=ZðGÞ
0
ð1ÞðjCjþjC jþjC
00
pTrs þ DZðGrs Þ=ZðGÞ and 1 þ DZðGÞ=ZðGÞ ~ ~pT DZðGrs Þ=ZðGÞ pTrs ¼ rs : ~ 1 DZðGÞ=ZðGÞ
~pTrs ¼
From these expressions and examples, one would be inclined to conclude that the connection between the topological bond order and its modified version is relatively complicated. However, our numerical testings show the opposite. These are outlined in the subsequent section.
jÞ=2
C;C 0 ;C 00
~ C C 0 C 00 Þ þ ZðG For instance, for the phenanthrene graph G and its subgraph G12 (see Fig. 1) we have:
3. Numerical work In Fig. 2 are plotted the topological bond orders of all bonds of the benzenoid hydrocarbons with 5 and
76
pT
I. Gutman et al. / Chemical Physics Letters 407 (2005) 73–77 0.34
pT
0.32 0.30 0.28 0.26 0.24 0.22 0.20 0.18 0.16 0.14 0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.30
0.32
fewer six-membered rings (a total of 25 molecules, 332 symmetry-non-equivalent bonds) versus the respective modified topological bond orders. From Fig. 2 is seen that a reasonably good, but far from perfect, linear correlation exists between the two bond orders. The respective regression formula is pTrs ¼ ð0:929 0:006Þ~pTrs þ ð0:036 0:001Þ; which is not very far from pTrs p~Trs . For non-benzenoid molecules (both alternant and non-alternant) the results are similar, yet somewhat more irregular. Details may be obtained from the authors (S.R.), upon request. As already mentioned, the relation between the Coulson and topological bond orders was much studied in the past [6,8]. In Fig. 3, we show such correlations for the carbon–carbon bonds of catacondensed benzenoid hydrocarbons. In these conjugated systems the carbon–carbon bonds can be classified into three groups, referred to as 3–3, 3–2, and 2–2. The vertices of the molecular graph, corresponding to an i–j bond have degrees equal to i and j. (Recall that the degree of a vertex is the number of its first neighbors. In benzenoid graphs only vertices of degree 2 and 3 occur.) From Fig. 3 three characteristic features are immediately recognized. First, there is a great deal of similarity between the diagrams (a) and (b), something that could be anticipated from the above discussed linear correlation between pT and ~ pT . Second, the correlation between T C p and p is of very poor quality; the same is true for the correlation between ~ pT and pC . Third, there is an obvious clustering of the data points with regards to above-described classification of the carbon–carbon bonds. This clustering is indicated by using different
(a)
0.40
pT Fig. 2. Correlation between the topological and modified topological bond orders, pertaining to benzenoid hydrocarbons with 5 and fewer six-membered rings. Empty and full circles correspond to catacondensed and pericondensed molecules, respectively. Correlation coefficient: 0.993.
0.32 0.31 0.30 0.29 0.28 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.19 0.18 0.17 0.16 0.15
pT
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.32 (b) 0.31 0.30 0.29 0.28 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.40 0.45
0.50
0.55
0.60
0.65
0.70
0.75
pC
0.80
0.80
pC Fig. 3. (a) Correlation between topological (pT) and Coulson bond orders (pC), pertaining to catacondensed benzenoid hydrocarbons with 5 and fewer six-membered rings. (b) The analogous correlation between modified topological ð~pT Þ and Coulson bond orders (pC). For the data points in diagrams (a) and (b) the correlation coefficients are r = 0.923 and r = 0.949, respectively. In spite of these apparently high r-values, it is more appropriate to speak about clustering of data points, than about their correlation. Triangles, circles and squares correspond to carbon–carbon bonds of type 3–3, 3–2, and 2–2, respectively; for details see text.
symbols (triangles, circles, squares) for 3–3, 3–2, and 2–2 bonds. Within each of these three clusters there is a further grouping of the data points, determined by the finer details of molecular topology. For instance, the data points pertaining to 2–2 bonds (squares) are separated into three groups, depending on the degrees of the first neighbors of the vertices participating in the respective bond; those belonging to the upper right group have first neighbors of degree 3; those in the lower left group have first neighbors of degree 2. The data points pertaining to 3–3 bonds (triangles) are separated into two groups, with the upper left group of points corresponding to bonds lying on the perimeter.
I. Gutman et al. / Chemical Physics Letters 407 (2005) 73–77
4. Concluding remarks The main conclusion resulting from of the present study is that instead of the topological bond order pT one may use its modified version ~ pT . Between these two bond-order-like quantities there is a relatively good linear correlation, and, if not a very high accuracy is required, one may accept that pTrs ~ pTrs . The numerical calculation of the modified topological bond order goes via the graph eigenvalues and is, thus, much easier than the calculation of pT. Therefore, in future applications of the topological-bond-order concept, one could safely employ its modified version. References [1] A. Graovac, I. Gutman, N. Trinajstic´, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin, 1977. [2] J.R. Dias, Molecular Orbital Calculations Using Chemical Graph Theory, Springer, Berlin, 1993.
77
[3] L. Pauling, L.O. Brockway, J.Y. Beach, J. Am. Chem. Soc. 57 (1935) 2705. [4] C.A. Coulson, Proc. Roy. Soc. Lond. A169 (1939) 413. [5] N.S. Ham, K. Ruedenberg, J. Chem. Phys. 29 (1958) 1215. [6] H. Hosoya, M. Murakami, Bull. Chem. Soc. Jpn. 48 (1975) 3512. [7] H. Hosoya, Bull. Chem. Soc. Jpn. 44 (1971) 2332. [8] H. Hosoya, K. Hosoi, J. Chem. Phys. 64 (1976) 1065. [9] H. Hosoya, J. Mol. Struct. (Theochem.) 461–462 (1999) 473. [10] H. Hosoya, Bull. Chem. Soc. Jpn. 76 (2003) 2233. [11] I. Gutman, MATCH Commun. Math. Comput. Chem. 6 (1979) 75. [12] I. Gutman, O.E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986. [13] D.M. Cvetkovic´, M. Doob, I. Gutman, A. Torgasˇev, Recent Results in the Theory of Graph Spectra, North-Holland, Amsterdam, 1988. [14] I. Gutman, MATCH Commun. Math. Comput. Chem. 3 (1977) 121. [15] I. Gutman, J. Serb. Chem. Soc. 43 (1978) 379. [16] H. Hosoya, K. Hosoi, I. Gutman, Theor. Chim. Acta 38 (1975) 37. [17] J. Aihara, J. Org. Chem. 41 (1976) 2488. [18] H. Hosoya, Theor. Chim. Acta 25 (1972) 215. [19] I. Gutman, O.E. Polansky, Theor. Chim. Acta 60 (1981) 203.
Some properties of the topological bond order Ivan Gutman a
a,*
, Slavko Radenkovic´ a, Boris Furtula a, Haruo Hosoya
b
Faculty of Science, University of Kragujevac, P.O. Box 60, YU-34000 Kragujevac, Serbia & Montenegro b Department of Information Sciences, Ochanomizu University, Bunkyo-ku, Tokyo 112–8610, Japan Received 4 February 2005; in final form 17 February 2005 Available online 7 April 2005
Abstract The topological bond order is a bond-order-like quantity, put forward in the 1970s. It is defined as pTrs ¼ ZðGrs Þ=ZðGÞ, where G is the molecular graph, Grs is obtained from G by deleting from it the adjacent vertices labelled by r and s, and Z stands for the respective topological (Hosoya) index. Because no easy way for the calculation of pTrs is known, its properties were studied only to a limited degree. We now introduce a modified topological bond order, ~ pTrs , that can (easily) be calculated from the eigenvalues of G and Grs. T T For acyclic systems, ~prs ¼ prs . In the case of polycyclic systems a reasonably accurate linear correlation exists between ~ pTrs and pTrs . Thus, by studying ~pTrs the main properties of pTrs can be established. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction In the topological theory of conjugated molecules [1,2] several bond-order-like quantities are considered. Of these, the best known are the Pauling et al. [3], the Coulson [4], and the Ham and Ruedenberg [5] bond orders. In the 1970s, one of the present authors proposed [6] the topological bond order, defined by means of the topological index.1 Thus, the topological bond order pertaining to the bond between atoms r and s is defined as pTrs ¼
ZðGrs Þ ; ZðGÞ
ð1Þ
where G is the molecular graph, Grs is the subgraph obtained from G by deleting from it the vertices corresponding to the atoms r and s, and Z stands for the
respective topological index. The topological index is defined as [7] X ZðGÞ ¼ mðG; kÞ; ð2Þ kP0
where m(G,0) = 1, m(G,1) = number of edges of the graph G, and m(G,k), k P 2, is the number of selections of k mutually independent (that is, mutually nontouching) edges in the graph G. Some fundamental properties of the topological bond order were established in two papers published in the 1970s [6,8], where also its relations to the Coulson and Pauling bond orders were analyzed. After that not much research in this area was done, as seen from the reviews [9,10]. An object, closely related to Z is the matching polynomial, defined as [11] X aðG; kÞ ¼ ð1Þk mðG; kÞkn2k ; ð3Þ kP0
*
Corresponding author. Fax: +38 134 335 040. E-mail address: [email protected] (I. Gutman). 1 The quantity Z, put forward in [7] and named Ôtopological indexÕ, is nowadays in the chemical literature usually referred to as the ÔHosoya indexÕ. We, nevertheless, call it by its original name. 0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.02.120
where n is the number of vertices of G. Details of the theory of the topological index and matching polynomial can be found in the books [12] and [13], respectively. Comparative studies of various bond orders
74
I. Gutman et al. / Chemical Physics Letters 407 (2005) 73–77
(including the topological bond order) were reported in [14,15]. Combining (2) and (3) one immediately obtains ð4Þ ZðGÞ ¼ in aðG; iÞ; pffiffiffiffiffiffiffi where i ¼ 1. Furthermore, if l1,l2, . . . ,ln are the zeros of the matching polynomial, then vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uY i u n h 2 ð5Þ 1 þ ðlj Þ : ZðGÞ ¼ t
Recall that DZ ¼ Z~ Z is considered as an Ôaromaticity indexÕ [10,16]. In non-alternant conjugated systems, the right-hand side of (8) may be complex-valued. Indeed, if we write the characteristic polynomial in the form X ak knk /ðG; kÞ ¼ kP0
then
j¼1
Formulas (4) and (5) are applicable to any (molecular) graph G. If G is acyclic, then the matching polynomial a(G,k) coincides with the characteristic polynomial / (G,k) [13], implying ZðGÞ ¼ in /ðG; iÞ
ð6Þ
and vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uY i u n h 2 ZðGÞ ¼ t 1 þ ðkj Þ ;
ð7Þ
j¼1
where k1,k2, . . . ,kn are the respective graph eigenvalues. In contrast to the zeros of the matching polynomial, graph eigenvalues are easily computed (using the powerful methods of linear algebra). Therefore, in contrast to (5), formula (7) provides a convenient route for the computation of the topological index. Combined with (1), formula (7) makes it possible to easily evaluate the topological bond orders of acyclic p-electron systems. Unfortunately, this graph-eigenvalue-based method for computing pTrs is not applicable in the case of cycle-containing conjugated molecules, which, from both experimental and theoretical points of view, are the most interesting and most frequently examined p-electron systems. In order to evade the computational difficulties encountered in the study of the topological bond orders of cyclic molecules, we consider an approximation whose essence is that the right-hand sides of Eqs. (6) and (7) are extended to all (molecular) graphs. The modified topological index Z~ thus obtained, namely ~ ZðGÞ ¼ in /ðG; iÞ
ð8Þ
was also investigated already in the 1970s [16], when also the expression vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uY i u n h 2 ~ ð9Þ ZðGÞ ¼t 1 þ ðkj Þ j¼1
~ rs Þ ZðG ~ ZðGÞ
was, so far, not investigated in due detail.
X X k k ð1Þ a2k i ð1Þ a2kþ1 kP0
ð10Þ
kP0
which is real-valued provided all odd coefficients (a2k + 1) are equal to zero. The fact that some odd coefficients of the characteristic polynomials of the molecular graphs of non-alternant hydrocarbons differ from zero, and that for alternant species all these coefficients are equal to zero, is a well-known consequence of the pairing theorem [1,2,12]. In what follows we restrict our considerations to alternant hydrocarbons, in which case this difficulty does not occur.
2. Relations between p~Trs and pTrs For acyclic systems, the topological bond order (1) and its modified version (10) coincide. In the case of polycyclic (molecular) graphs the relation between the two bond orders is much more complicated. Let G be a graph and C, C 0 , C 0 0 , . . . be its cycles. Let |C| denote the size of the cycle C and let G C be the subgraph obtained from G by deleting from it all vertices of C. Two cycles C and C 0 are said to be disjoint if they have no common vertex. For an example see Fig. 1. Let, as before, /(G) = /(G,k) and a(G) = a(G,k) denote the characteristic and matching polynomials of G. Then [13,18,19]: /ðGÞ ¼ aðGÞ 2 8
X
aðG CÞ þ 4
aðGÞ ¼ /ðGÞ þ 2
aðG C C 0 Þ
C;C 0
C
X
X
aðG C C 0 C 00 Þ þ
C;C 0 ;C 00
þ8
was established [17]. However, the analogous modified topological bond order, viz. ~ pTrs ¼
in /ðG; iÞ ¼
X
X
/ðG CÞ þ 4
C
X C;C
/ðG C C 0 Þ
0
0
/ðG C C C 00 Þ þ
C;C 0 ;C 00
where the summations go over all cycles, over all pairs of disjoint cycles, over all triples of mutually disjoint cycles, etc., of the graph G. In the case of alternant hydrocarbons, when all |C|-values are even, by means of (4) and (8) we get:
I. Gutman et al. / Chemical Physics Letters 407 (2005) 73–77
~ ZðGÞ¼ZðGÞþ2½ZðGC 1 ÞþZðGC 2 ÞþZðGC 3 Þ þZðGC 4 ÞþZðGC 5 ÞþZðGC 6 Þ
6 2 3
1
5
7
4
8
þ4ZðGC 1 C 3 Þ; ~ ZðG12 Þ¼ZðG12 Þþ2½ZðG12 C 2 ÞþZðG12 C 3 ÞþZðG12 C 5 Þ; ~ ~ ~ ~ ZðGÞ¼ ZðGÞ2½ ZðGC 1 Þþ ZðGC 2 Þþ ZðGC 3 Þ
9
14
10
12 11
13
75
G
G12
C1
C2
~ ~ ~ þ ZðGC 4 Þþ ZðGC 5 Þþ ZðGC 6 Þ ~ þ4ZðGC 1 C 3 Þ; ~ 12 Þ2 ZðG ~ 12 C 2 Þþ ZðG ~ 12 C 3 Þþ ZðG ~ 12 C 5 Þ : ZðG12 Þ¼ ZðG
Bearing in mind that Z(G C1) = Z(G C3) = 44, Z(G C2) = 25, Z(G C4) = Z(G C5) = 5, Z(G C6) = 1, Z(G C1 C3) = 2, and Z(G12 C2) = 10, Z(G12 C3) = 12, Z(G C5) = 2, we get
C3
~pT12 ¼
C4
¼
C6
C5
ZðG12 Þ þ 2ð10 þ 12 þ 2Þ ZðGÞ þ 2ð44 þ 25 þ 44 þ 5 þ 5 þ 1Þ þ 4ð2Þ ZðG12 Þ þ 48 ; ZðGÞ þ 256
i.e. ~pT12 ¼
G-C2
G-C1
G12-C3
G12-C5
Fig. 1. The molecular graph G of phenanthrene and of its vertexdeleted subgraph G12. The cycles contained in G are C1,C2, . . . ,C6 and are indicated by heavy lines. The only two disjoint cycles in G are C1 and C3. The graph G does not possess triples, quadruples, etc., of mutually disjoint cycles. The subgraph G12 contains only the cycles C2, C3, C5. Some typical subgraphs of the form G C and Grs C are also shown.
pT12 þ 48=ZðGÞ : 1 þ 256=ZðGÞ
~ C 1 Þ ¼ ZðG ~ C 3 Þ ¼ 48, Bearing in mind that ZðG ~ ~ ~ ZðG C 2 Þ ¼ 25, ZðG C 4 Þ ¼ ZðG C 5 Þ ¼ 5, ZðG ~ ~ C 6 Þ ¼ 1, ZðG C 1 C 3 Þ ¼ 2, and ZðG12 C 2 Þ ¼ 10, ~ C 3 Þ ¼ 12, ZðG ~ C 5 Þ ¼ 2, we get ZðG ~ 12 Þ 2ð10 þ 12 þ 2Þ ZðG ~ ZðGÞ 2ð48 þ 25 þ 48 þ 5 þ 5 þ 1Þ þ 4ð2Þ ~ 12 Þ 48 ZðG ¼ ; ~ ZðGÞ 256
pT12 ¼
i.e. ~ ZðGÞ ¼ ZðGÞ 2 þ4
X
X
ð1Þ
jCj=2
pT12 ¼ ZðG CÞ
C
ð1Þ
In the general case ðjCjþjC 0 jÞ=2
ZðG C C 0 Þ
C;C 0
8
X
0
ð1ÞðjCjþjC jþjC
00
jÞ=2
C;C 0 ;C 00
ZðG C C 0 C 00 Þ þ X jCj=2 ~ ~ ZðGÞ ¼ ZðGÞ þ2 ZðG CÞ ð1Þ þ4
X
C 0
~ C C0Þ ð1ÞðjCjþjC jÞ=2 ZðG
C;C 0
þ8
X
~ ~pT12 48=ZðGÞ : ~ 1 256=ZðGÞ
0
ð1ÞðjCjþjC jþjC
00
pTrs þ DZðGrs Þ=ZðGÞ and 1 þ DZðGÞ=ZðGÞ ~ ~pT DZðGrs Þ=ZðGÞ pTrs ¼ rs : ~ 1 DZðGÞ=ZðGÞ
~pTrs ¼
From these expressions and examples, one would be inclined to conclude that the connection between the topological bond order and its modified version is relatively complicated. However, our numerical testings show the opposite. These are outlined in the subsequent section.
jÞ=2
C;C 0 ;C 00
~ C C 0 C 00 Þ þ ZðG For instance, for the phenanthrene graph G and its subgraph G12 (see Fig. 1) we have:
3. Numerical work In Fig. 2 are plotted the topological bond orders of all bonds of the benzenoid hydrocarbons with 5 and
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pT
I. Gutman et al. / Chemical Physics Letters 407 (2005) 73–77 0.34
pT
0.32 0.30 0.28 0.26 0.24 0.22 0.20 0.18 0.16 0.14 0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.30
0.32
fewer six-membered rings (a total of 25 molecules, 332 symmetry-non-equivalent bonds) versus the respective modified topological bond orders. From Fig. 2 is seen that a reasonably good, but far from perfect, linear correlation exists between the two bond orders. The respective regression formula is pTrs ¼ ð0:929 0:006Þ~pTrs þ ð0:036 0:001Þ; which is not very far from pTrs p~Trs . For non-benzenoid molecules (both alternant and non-alternant) the results are similar, yet somewhat more irregular. Details may be obtained from the authors (S.R.), upon request. As already mentioned, the relation between the Coulson and topological bond orders was much studied in the past [6,8]. In Fig. 3, we show such correlations for the carbon–carbon bonds of catacondensed benzenoid hydrocarbons. In these conjugated systems the carbon–carbon bonds can be classified into three groups, referred to as 3–3, 3–2, and 2–2. The vertices of the molecular graph, corresponding to an i–j bond have degrees equal to i and j. (Recall that the degree of a vertex is the number of its first neighbors. In benzenoid graphs only vertices of degree 2 and 3 occur.) From Fig. 3 three characteristic features are immediately recognized. First, there is a great deal of similarity between the diagrams (a) and (b), something that could be anticipated from the above discussed linear correlation between pT and ~ pT . Second, the correlation between T C p and p is of very poor quality; the same is true for the correlation between ~ pT and pC . Third, there is an obvious clustering of the data points with regards to above-described classification of the carbon–carbon bonds. This clustering is indicated by using different
(a)
0.40
pT Fig. 2. Correlation between the topological and modified topological bond orders, pertaining to benzenoid hydrocarbons with 5 and fewer six-membered rings. Empty and full circles correspond to catacondensed and pericondensed molecules, respectively. Correlation coefficient: 0.993.
0.32 0.31 0.30 0.29 0.28 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.19 0.18 0.17 0.16 0.15
pT
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.32 (b) 0.31 0.30 0.29 0.28 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.40 0.45
0.50
0.55
0.60
0.65
0.70
0.75
pC
0.80
0.80
pC Fig. 3. (a) Correlation between topological (pT) and Coulson bond orders (pC), pertaining to catacondensed benzenoid hydrocarbons with 5 and fewer six-membered rings. (b) The analogous correlation between modified topological ð~pT Þ and Coulson bond orders (pC). For the data points in diagrams (a) and (b) the correlation coefficients are r = 0.923 and r = 0.949, respectively. In spite of these apparently high r-values, it is more appropriate to speak about clustering of data points, than about their correlation. Triangles, circles and squares correspond to carbon–carbon bonds of type 3–3, 3–2, and 2–2, respectively; for details see text.
symbols (triangles, circles, squares) for 3–3, 3–2, and 2–2 bonds. Within each of these three clusters there is a further grouping of the data points, determined by the finer details of molecular topology. For instance, the data points pertaining to 2–2 bonds (squares) are separated into three groups, depending on the degrees of the first neighbors of the vertices participating in the respective bond; those belonging to the upper right group have first neighbors of degree 3; those in the lower left group have first neighbors of degree 2. The data points pertaining to 3–3 bonds (triangles) are separated into two groups, with the upper left group of points corresponding to bonds lying on the perimeter.
I. Gutman et al. / Chemical Physics Letters 407 (2005) 73–77
4. Concluding remarks The main conclusion resulting from of the present study is that instead of the topological bond order pT one may use its modified version ~ pT . Between these two bond-order-like quantities there is a relatively good linear correlation, and, if not a very high accuracy is required, one may accept that pTrs ~ pTrs . The numerical calculation of the modified topological bond order goes via the graph eigenvalues and is, thus, much easier than the calculation of pT. Therefore, in future applications of the topological-bond-order concept, one could safely employ its modified version. References [1] A. Graovac, I. Gutman, N. Trinajstic´, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin, 1977. [2] J.R. Dias, Molecular Orbital Calculations Using Chemical Graph Theory, Springer, Berlin, 1993.
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[3] L. Pauling, L.O. Brockway, J.Y. Beach, J. Am. Chem. Soc. 57 (1935) 2705. [4] C.A. Coulson, Proc. Roy. Soc. Lond. A169 (1939) 413. [5] N.S. Ham, K. Ruedenberg, J. Chem. Phys. 29 (1958) 1215. [6] H. Hosoya, M. Murakami, Bull. Chem. Soc. Jpn. 48 (1975) 3512. [7] H. Hosoya, Bull. Chem. Soc. Jpn. 44 (1971) 2332. [8] H. Hosoya, K. Hosoi, J. Chem. Phys. 64 (1976) 1065. [9] H. Hosoya, J. Mol. Struct. (Theochem.) 461–462 (1999) 473. [10] H. Hosoya, Bull. Chem. Soc. Jpn. 76 (2003) 2233. [11] I. Gutman, MATCH Commun. Math. Comput. Chem. 6 (1979) 75. [12] I. Gutman, O.E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986. [13] D.M. Cvetkovic´, M. Doob, I. Gutman, A. Torgasˇev, Recent Results in the Theory of Graph Spectra, North-Holland, Amsterdam, 1988. [14] I. Gutman, MATCH Commun. Math. Comput. Chem. 3 (1977) 121. [15] I. Gutman, J. Serb. Chem. Soc. 43 (1978) 379. [16] H. Hosoya, K. Hosoi, I. Gutman, Theor. Chim. Acta 38 (1975) 37. [17] J. Aihara, J. Org. Chem. 41 (1976) 2488. [18] H. Hosoya, Theor. Chim. Acta 25 (1972) 215. [19] I. Gutman, O.E. Polansky, Theor. Chim. Acta 60 (1981) 203.